Forecast Accuracy Improvement:
Evidence from U.S. Nonfarm Payroll Employment
Allan W. Gregory, Hui (Julia) Zhu
∗
Department of Economics, Queen’s University†
May 2008
Abstract
The timing of data release for time series variables for the same time period of observation is
often spread over weeks. For instance official government statistics are often released at different
times over the quarter or month and yet cover the same time period. This paper focuses on this
separation of announcement time or data release and the use of standard econometric updating
methods to forecast data that has yet to be made available. The theoretical framework shows that
the updating root mean squared forecast error will be accurate with higher correlation coefficients
among observation innovations. There are large amounts of applications to earnings forecasts for
firms and macroeconomic forecasting. With fixed timing announcement, we find that data on U.S.
nonfarm payroll employment are consistent with the theoretical framework. The results highlight
an important multivariate forecast accuracy measure.
∗
The author is grateful to and for helpful comments and suggestions. She also thanks seminar participants at
Department of Economics, Queen’s University.
†
Address correspondence to Hui (Julia) Zhu, Department of Economics, Queen’s University, Kingston, Ontario,
Canada K7L 3N6; E-mail: hzhu@econ.queensu.ca.
1
Introduction
The timing of data release for time series variables for the same time period of observation
is often spread over weeks. For instance earning announcements for firms can be spread over a
two week period even though these earnings are for the same quarter or month. Further, official
government statistics are often released at different times over the quarter or month and yet cover
the same time period. This paper focuses on this separation of announcement time or data release
and the use of standard econometric updating methods to forecast data that has yet to be made
available. To the best of our knowledge this rather important aspect of forecasting which we think
has direct application in financial market updating (in terms of earnings, earning per share and so
on) has not been studies in the literature.
Traditional forecasting in a multivariate time series setting usually is studied in the context of
vector autoregression (VAR) models. In this set-up, some common end-point, the VAR is specified,
estimated and single or multiple period forecasts are conducted. Standard errors for the forecasts
can be based on asymptotic normal theory or more recently use of bootstrap or some other resampling technique can be applied. The key element for this exercise is that there is a common end
point of observed data and that forecasts are made for all variables over some common forecasting
horizon.
The situation that we consider is different in that only some (one) of the variables comprising
the system is released at a point in time with the remaining variables are release at latter dates.
The latter dates may coincide for some variables or it may differ in which case there is a sequence
of release times for the system as a whole.
There are two critical exogeniety assumptions in this exercise. The first assumption that the
timing of the release information (say earlier or later in the release cycle) does not depend on the
information that is released. That is, if firms have poor earnings they might choose to get this
information out to prospective investors early (or later). From what we are able to tell about
earnings announcements, the decisions as to when to announce is made long in advance of when
the earnings information would be credibly known to the firm so this is unlikely to be a problem.
The second exogeniety assumption is that there is no feedback on one firm’s earnings with those of
another. That is, if one firm announces large earnings in say Q4 for 2007 in February 27 2008, that
a related firm does not change its announced earnings for the same quarter which is released on
say February 28 2008. For the present application we avoid both of these issues and start a simple
bivariate example using nonfarm labor data in the United States.
Comparison with the ordinary VAR forecasts shows that the updating forecast mean squared
error (MSE) are smaller than the one in the ordinary VAR forecast. We first consider one realtime variable available case in the bivariate updating forecasts. When the correlation coefficient
of the innovations between two variables are lower, the updating forecast has the relative smaller
MSE than the ordinary VAR forecast. When the correlation coefficient of observation innovations
between two variables are higher, the updating forecast has a significant smaller MSE than the
ordinary VAR forecast. In one extreme case, when correlation coefficient approaches to zero, that
1
is, no contemporaneous information is available to be useful for forecasting, the updating forecast
has the exact same result as the ordinary VAR forecast. In another extreme case, when correlation
coefficient approaches to one, that is, we have perfect linear association and there is no errors, the
updating forecast is the best. Furthermore, we note that not only the correlation coefficient of
observation innovation but also the coefficient parameter plays an important role in the multistep
ahead forecast. Moreover, this research is the most up-to-date forecasting. To solve data revision
problem in the sample, whenever data is available at the time period of forecasting, we adopt
it. Finally, Application to nonfarm labor data in the United States shows that the root mean
squared error in the updating VAR is less than the ordinary VAR and univariate autoregression.
It shows quantitatively that the updating VAR improves 17% in one-step ahead forecast and 12%
in two-stepa ahead forecast respectively comparing to the ordinary VAR.
Related Literature
After Sims’s (1980) influential work, VAR are widely applied in analyzing the dynamics of economic system. For over twenty years, multivariate VAR model has been proven to be powerful and
reliable tools in everyday use. Stock and Watson (2001) reassess how well VARs have addressed in
data description, forecasting, structural inference, and policy analysis. Working with the inflationunemployment-interest rate VAR, they conclude that VAR either does no worse than or improves
upon the univariate autoregression and that both improve upon the random walk forecast. Therefore, VAR model is now rightly in data description and forecasting. However, the standard VAR
forecast does not match the facts that the timing of data release for most financial time series and
macro time series for the same time period of observation is often spread over weeks.
Many economic time series are subject to revision. Revisions to measures of real economic
activity may occur immediate consequent month or years after official figures are first released.
Aruoba (2008) documents the empirical properties of revisions to major macroeconomic variables
in the United States. He finds that there revision do not have zero mean, which indicates that the
initial announcements by Statistical Agencies are biased. Croushore and Stark (2001, 2003) show
how data revisions can affect forecast. They use a real-time data set to analyze data revisions.
While the results of some studies are conflicting, Koenig et al. (2003) show first-release data are
to be preferred for estimation even if the analyst is ultimately interest in predicting revised data.
They provide three alternative strategies for estimating forecasting equations with real time data.
They conclude that using first released data in the both sides of equation performs superior forecast
to that obtaining final released data. This paper employs the most up-to-date estimates available
at forecasting time in addition to the updating VAR forecast method.
The primary use of earnings or macroeconomic forecasts is to provide a proxy for the market
expectation of a future realization. Recent work suggests that the stock market reacts to earnings
or macroeconomic announcements. Some researches study how markets respond to labor data, such
as Krueger (1996) and Boyd et al. (2005). The latter examines how stocks respond to unemployment news, which is measured as the surprise component. Adopting forecasts of the change in the
unemployment rate to get the surprise component in the announcement of the unemployment rate,
2
they find that an announcement of rising unemployment is good news for stocks during economic
expansions and bad news during economic contractions. Another line of researches focus on how
markets respond to macro variables, such as Flannery et al. (2002), Pesaran and Wickens (1995),
and Rapach et al. (2005). Faust et al. (2007) studies how U.S. macroeconomic announcements
affect joint movements of exchange rates and interest rates. Using 14-year span of high-frequency
data, they conclude that unexpectedly strong announcements lead either to a fall in the risk premium required for holding foreign assets or an expected net depreciation over the ensuing decade,
or both.
Role of Current Research
The existing literature on accuracy of time series forecasts has focused on fixed time period.
With revision in data, most estimation adopts final released data (end-of-sample). This research
adds several new dimensions. First, we provide a practical updating multivariate VAR forecast
method, extending the analysis into U.S. nonfarm payroll employment. Second, we construct the
most up-to-date data set for estimation and forecast basis. In this paper, first released data plays
an important role in estimation and forecast. Finally, we investigate how stock market reacts to
current employment situation, which is the first major economic indicator released each month.
The remainder of the paper is organized as follows. Section 2 derives multistep ahead bivariate
updating VAR forecasts based on one real-time variable available. In section 3 we discuss the
model specification. An application to U.S. nonfarm payroll employment is illustrated in section 4.
Section 5 concludes. We collect proofs in the Appendix.
2
A Theoretical Framework
Consider the N multivariate stationary VAR (1) model
Yt = AYt−1 + ²t ,
t = 0, ±1, ±2, ...,
(1)
where Yt = (y1t , ..., yN t )0 is a (N × 1) random vector, the A is fixed (N × N ) coefficient matrices.
Moreover, ²t = (²1t , ..., ²N t )’ is a N -dimensional white noise or innovation process, that is, E(²t ) = 0
and E(²t ²0t ) = Ω, with the contemporaneous covariance C(²it , ²jt ) = ρij σi σj for i = 1, ..., N and
j = 1, ..., N . The covariance matrix Ω is assumed to be nonsingular if not otherwise stated. Since
we assume stationary vector autoregressive process, the condition of correlation coefficient |ρij | < 1
must hold. Finally, σi is the standard deviation of the innovation ²i . Given the multivariate VAR
model (1), the ordinary VAR one-step ahead forecast at time region T is YT (1) = A YT and the
associated forecast error and the corresponding MSE or the covariance matrix of forecast error are
²T (1) = ²T +1 and M SEY is in-diagonal covariance matrix of ΩI 0 respectively, where YT (1) denotes
the forecast of Y at time T + 1, ²T (1) denotes forecast error at time T + 1, and I is (N × N )
identity matrices. The ordinary multivariate VAR forecasts are standard and can be obtained from
Lütkepohl (1993) and Hamilton (1994).
3
Theoretically, multivariate VAR forecast is based on the certain amount of time periods of 1
through T . Each equation in multivariate VAR model can be estimated by ordinary least squares
(OLS) regression. This OLS estimator is as efficient as maximum likelihood estimator and general
least squared estimator. Therefore, the ordinary VAR forecast computed through the unbiased and
consistent coefficient estimates and the variance covariance matrix estimates has the lowest MSE
and is optimal. However, the fact is that most macroeconomic or financial time series we study
in multivariate VAR forecast are not ending at the same time, that is, one variable is generally
available for public use a couple days prior to the other variables. For example, financial time series
such as firms’ earnings releases for the same quarter or the same year are available at different dates
for public use; macro indicators such as inflation, employment, unemployment rate, interest rate,
and etc. releases for the same month are also available at different dates for public use. Omitting
the timing fact, the ordinary multivariate VAR forecast usually ignores the latest information we
can obtain and adopts the same amount of certain time periods to do forecasts.
The focus of this paper is on examining how taking advantage of one or more information about
one variable and matching the timing fact to improve multivariate VAR forecasts.
2.1
Bivariate VAR with Multistep Ahead Forecast
A practical method to update forecasts with one real-time variable available in advance is
investigated. To simplify the discussion, consider N = 2, y1t be observable for t = 1, . . . T + 1, and
y2t be observable only up to time T . Then the reduced form bivariate VAR (1) model is as follows.
y1t = a11 y1t−1 + a12 y2t−1 + ²1t ,
t = 1, ..., T, T + 1
y2t = a21 y1t−1 + a22 y2t−1 + ²2t ,
t = 1, ..., T
(2)
Following the ordinary VAR forecasting method proposed by Lütkepohl (1993), the one-step
ahead ordinary VAR forecast error covariance matrix (or forecast MSE matrix) is
M SE[YT (1)] = A0 Ω² (A0 )0
where Y is a vector of (y1 , y2 )0 , A is 2 × 2 dimensional coefficient matrix, and Ω² is covariance
matrix of E(²t , ²0t ). That is
Ã
A=
and
Ã
Ω² =
a11 a12
!
,
a21 a22
σ12
ρ12 σ1 σ2
ρ12 σ1 σ2
σ22
!
.
With one more real-time information available in advance, the k = 1 horizon forecast is performed in the following. Thereafter we denote MSEu as the updating VAR forecast MSE to distinguish with the ordinary VAR forecast MSE.
4
Proposition 1 Given the known full information set I = {y11 , y12 , ..., y1T +1 , y21 , ..., y2T }, the mean
squared error of one-step ahead forecast of variable y2 in the updating bivariate VAR is
M SE u [y2T (1)] = V ar[y2T +1 − ŷ2T +1 ] = V ar[²2T − ²2T (1)] = (1 − ρ212 )σ22
(3)
Proof. See Appendix.
This proposition shows that taking advantage of one more real-time information available in
advance, the one-step ahead updating bivariate VAR forecast MSEu is (1 − ρ212 )σ22 . Since ρ12
is coefficient correlation between ²1 and ²2 , the condition of ρ12 < 1 must hold by assumption
of stationary vector autoregressive process. However, omitting the extra information y1T +1 , the
ordinary bivariate VAR forecast MSE gives M SE = σ22 . Therefore, the one-step ahead updating
bivariate VAR forecast has smaller MSE comparing with the forecast from the ordinary bivariate
VAR. This implies the updating bivariate VAR forecast is more accurate than the forecast from the
ordinary bivariate VAR. Additionally, the higher correlation among the error terms of the variables,
the smaller MSE in the updating VAR forecast.
With one more real-time information available in advance, the k > 1 long-horizon forecast is
also examined.
Proposition 2 Given the known full information set I = {y11 , y12 , ..., y1T +1 , y21 , ..., y2T }, the kstep ahead forecast mean squared error matrix in the updating bivariate VAR is
Ã
M SE u [YT (k)] =
0
0
!
i
i 0
+ Σk−1
i=1 A Ω² (A ) ,
0 (1 − ρ212 )σ22
k > 1.
(4)
Proof. See Appendix.
This proposition shows that multistep ahead updating bivariate VAR forecast takes advantage
of the first step forecast derived from proposition 1. By iterating forward, the k-step ahead updating bivariate VAR forecast mean squared error matrix MSEu in equation (4) is smaller than the
recursive ordinary bivariate VAR forecast MSE matrix
M SE[YT (k)] = Σk−1
Ai Ω² (Ai )0
!
Ãi=0
σ12
ρ12 σ1 σ2
i
i 0
+ Σk−1
=
i=1 A Ω² (A ) ,
2
ρ12 σ1 σ2
σ2
k > 1.
To see this, the difference between the updating VAR forecast MSEu and ordinary VAR forecasts
5
MSE is as follows.
Ã
i
i 0
M SE[YT (k)] − M SE u (YT +k ) = Σk−1
i=0 A Ω² (A ) −
Ã
= Ω² −
Ã
=
0
!
0
0 ρ212 σ22
σ12
ρ12 σ1 σ2
ρ12 σ1 σ2
σ22
0
!
0
0 ρ212 σ22
!
Ã
−
i
i 0
− Σk−1
i=1 A Ω² (A )
0
0
!
0 (1 − ρ212 )σ22
> 0
Therefore, the multistep ahead updating bivariate VAR forecast has smaller MSE comparing with
the forecast from the ordinary bivariate VAR. This implies the updating bivariate VAR forecast
is more accurate than the forecast from the ordinary bivariate VAR. Additionally, the higher
correlation among the error terms of the variables, the smaller MSE in the updating VAR forecast.
3
Model Specification
There are alternative ways to model the relationship between two time series under the as-
sumption in section 2. The contemporaneous regression is the traditional approach to study. In
the following two subsection we show that the linear regression model is misspecified and that
bivariate VAR model gains in efficiency.
To investigate misspecification of contemporaneous regression, consider a reduced form bivariate
VAR (1) process,
y1t = a11 y1t−1 + a12 y2t−1 + ²1t
(5)
y2t = a21 y1t−1 + a22 y2t−1 + ²2t
(6)
We see from (5) that y2t−1 = (y1t − a11 y1t−1 − ²1t )/a12 . If we substitute this into (6), we find that
y2t =
a22
a11 a22
a22
y1t + (a21 −
)y1t−1 + ²2t −
²1t
a12
a12
a12
(7)
Since y1t is correlated to the error term ²1t by assumption, this expression (7) is equivalent to
instrument variables estimation as follows:
y2t =
a22 u
a11 a22
y
+ (a21 −
)y1t−1 + ²2t
a12 2t−1
a12
(8)
u
where the variables y2t−1
is not actually observed. Instead, we observe
u
y1t = y2t−1
+ ²1t .
(9)
Here ²1t is measurement error which is assumed to be identified, independent, and distributed with
6
variance σ12 and to be independent of y1t−1 and y2t . It is also assumed that there is contemporaneous
correlation of ²1t and ²2t . Therefore, E(²1t ²2t ) = ρ12 for some correlation coefficient ρ12 such that
−1 < ρ12 < 1.
If we suppose that the true DGP is the VAR (1) model, that is a special case of (8) along with
equation (9), from (7) we find that
a11 a22
a22
a22
y1t + (a21 −
)y1t−1 + ²2t −
²1t
a12
a12
a12
= αy1t−1 + βy1t + ut
y2t =
(10)
where
α ≡ a21 −
a11 a22
,
a12
β≡
a22
,
a12
ut ≡ ²2t − β²1t
Thus V ar(ut ) is equal to σ22 + β 2 σ12 − 2βρσ1 σ2 .
If we estimate simple model like (10), one effect of the measurement error in the independent
variable is to increase the variance of the error terms if ²1 and ²2 are negatively correlated. Another
u
severe consequence is that the OLS estimator is biased and inconsistent. Because y1t = y2t−1
+ ²1t ,
and ut depends on ²1t , ut must be correlated with y1t whenever β 6= 0. In fact, since the random
part of y1t is ²1t and ²1t is correlated to ²2t , we have that
E(ut |y1t ) = E(ut |²1t ) = ²2t − β²1t .
Using the fact that E(ut ) = 0 unconditionally, we can see that
Cov(y1t , ut ) = E(y1t ut )
= E(y1t E(ut |y1t ))
= E((y2t−1 + ²1t )(²2t − β²1t ))
= ρσ1 σ2 − βσ12
This covariance is negative if β > ρσ2 /σ1 and positive if β < ρσ2 /σ1 . Since it does not depend
on the sample size n, it does not go away as n becomes large. Therefore the OLS assumption
that E(ut |Xt ) = 0 is false whenever any element of Xt , that is y1t and y1t−1 in our special case, is
measured with error. In consequence, the OLS estimator is biased and inconsistent.
Proposition 3 Given the true DGP of a reduced form VAR (1) in equations (5) and (6), the OLS
estimator in the linear regression model (10) is biased and inconsistent.
Proof. See Appendix.
This proposition shows when the true DGP is a reduced form bivariate VAR (1), linear regression
model is misspecified and OLS estimators lead serious measurement error problem.
7
4
Application to U.S. Nonfarm Payroll Employment
There are a large amount of applications of the updating bivariate VAR forecast. We focus on
examining one of the macroeconomic indicators such as employment in the United States.
Bureau of Labor Statistics (BLS), U.S. Department of Labor, releases the employment situation
each month. The announcements are usually made at 8:30 a.m. on a Friday. The employment situation is composed of household survey and establishment survey data. The household survey has a
wider scope than the establishment survey since it includes the self-employed, unpaid family workers, agricultural workers, and private household workers, who are excluded by the establishment
survey. However, the establishment survey employment series has a smaller margin of error on the
measurement of month-to-month change than the household survey in that it has much larger sample size. The establishment survey includes several industry payroll employment information, such
that the total nonfarm payroll employment, weekly and hourly earnings, weekly hours worked, and
by selected industry. As a set of labor market indicators, nonfarm payroll employment counts the
number of paid employees working part-time or full-time in the nation’s business and government
establishments. They are used in the formulation of fiscal and economic policy. Federal officials
constantly monitor this data watching for even smallest signs of potential inflationary pressure.
The series are also used as an explanation of anomalous stock price behavior. By tracking the jobs
data, investors can sense the degree of tightness in the job market, which affects interest rates and
leads bond and stock prices fluctuate in the process.
Automatic Data Processing (ADP) contracted with Macroeconomic Advisors to compute a
monthly report (ADP National Employment Report). Estimates of employment published in the
ADP National Employment Report are available beginning in January of 2001. The announcements
are usually made at 8:15 a.m. on a Wednesday, although very less announcements are made on
other days. All announcement dates, whether Wednesday or not, are included in our study. The
ADP report is a measure of nonfarm private employment, and it forms the level of employment
by select industry and by size of payroll (small, medium, and large). It ultimately helps to predict
monthly nonfarm payrolls from BLS employment situation. The ADP report only covers private
payrolls, excluding government. However, it has been proved that nonfarm private employment
released on the behalf of the ADP (hereafter called ADP’s data) are highly correlated to nonfarm
payroll employment announced two days later by BLS (hereafter called BLS’s data). As a result,
this study only focuses on ADP’s data and BLS’s data.
In this section, data description is illustrated firstly. Then we apply ADP’s data and BLS’s data
to fit the bivariate updating VAR model and study VAR forecast performance. Comparisons to
the various individual time series forecasts are also investigated. Afterall, impacts on stock market
are studied.
8
Table 1: Descriptive Statistics
Source
Frequency
Sample period
Sample Size
Mean
Standard deviation
β1
Augmented Dickey-Fuller
H0 : Nonstationary
Automatic Data Processing, Inc.
Macroeconomic Advisers, LLC.B
Bureau of Labour Statistics
Seasonal adjusted, Monthly
January 2001 - April 2008
88
First difference of adp
49.0814
134.0006
-0.1154
-2.273
Reject at 90% (critical value = -1.61)
First difference of bls
63.2674
155.1345
-0.2629
-3.556
†
The coefficient β designates the autocorrelation of the series at lag i. The augmented Dickey-Fuller test statistic is
computed as τ̂ = β̂/ase(β̂) in the model: ∆yt = Σpj=1 βp yt−p + εt , with p = 1 lags.
4.1
Data
Two time series, namely ADP’s data adp and BLS’s data bls, are used to fit the bivariate VAR
model in (1). bls is total nonfarm payroll employment from nonfarm payrolls establishment survey
conducted by state employment security agencies in cooperation with the BLS. adp is nonfarm
private employment from the ADP national employment report computed by the ADP on behalf
of Macroeconomic Advisers. The 88 observations are seasonal adjusted monthly from January of
2001 to April of 2008. Table 1 reports the descriptive statistics. Dickey-Fuller nonstationary tests
have been conducted, and the presence of a unit root is rejected. Both series are stationary use first
differences. Since the test is known to have low power, even a slight rejection means that existence
of a unit root is unlikely. The time-series plot of the data is provided in Figure 1. After the first
difference of each time-series, the plot of the monthly changes of employees on nonfarm payrolls is
provides in Figure 2.
The timing of two time series release is fixed and in order. The ADP national employment
report is released, for public use only, two days prior to publication of the employment situation
by the BLS. For year 2008 release schedule, August 2008 report is the only one which is release for
public use only one day prior to publication of the employment situation by the BLS.
The BLS revises its initial monthly estimates twice, in the immediately succeeding 2 months,
to incorporate additional information that was not available at the time of the initial publication of
the estimates. On an annual basis, the BLS re-calculates estimates to nearly complete employment
counts available from unemployment insurance tax records, usually in March. ADP revises its
initial release one month later. In addition, the entire history of the report is revised once a year
after the annual benchmarking of the national employment data by the BLS, usually in February.
9
Figure 2: Changes in Nonfarm Payrolls
2000m1
−400
110000
−200
120000
0
130000
200
400
140000
Figure 1: Nonfarm Payrolls in Level
2002m1
2004m1
Month
ADP nonfarm private in level
2006m1
2008m1
2001m1
BLS nonfarm payrolls in level
2002m1
2003m1
2004m1 2005m1
Month
Changes of level in ADP
2006m1
2007m1
2008m1
Changes of level in BLS
Source: Automatic Data Processing, Inc., Macroeconomic Advisers, LLC., Bureau of Labour Statistics.
Nonfarm payroll employment counts the number of paid employees working part-time or full-time in the nation’s
business and government establishments. The ADP national employment report only covers private payrolls,
excluding government.
We incorporate the features of revisions existing in bls and adp time series into time series model
estimation and forecasting.
4.2
Forecasting
A reduced form VAR expresses each variable as a linear function of its own past values, the
past values of all other variables being considered. It also captures a serially correlated error term
across equations. The error terms in these regressions are the co-movements in the variables after
taking its past values into account. Thus, in this study the VAR involves two equations: current
adp as a function of past values of the adp and the bls; current bls as a function of past values of
the adp and the bls. Each equation can be estimated by ordinary least squares regression. This
OLS estimator is as efficient as maximum likelihood estimator and general least squared estimator.
The number of lagged values to include in each equation is determined by Schwarz’s Bayesian
information criterion (SBIC), Akaike’s information criterion (AIC) and the Hannan and Quinn
information criterion (HQIC). All latter two criterions indicate that the optimal lag selection of
two combining two time series and the optimal lag selection of two for each individual time series.
SBIC suggests the optimal lag selection of one for either combining two time series or individual
time series. For comparison we employ one lag in all time series models such as the updating VAR,
the ordinary VAR, and univariate autoregression.
The bivariate VAR (1) model estimation with standard deviation under the coefficients is as
follows:
adpt = 0.7882 adpt−1 + 0.0975 blst−1 + ²1t
(0.0930)
(0.0793)
blst = 0.8388 adpt−1 + 0.1342 blst−1 + ²2t
(0.1319)
(0.1125)
10
In the ordinary VAR forecast, the coefficient estimates and the variance covariance matrix estimates are used to calculate the forecast mean squared error. In the updating VAR forecast,
in addition to coefficient estimates and variance covariance matrix estimates, we also concern the
correlation of residuals of VAR to compute the more efficient forecast mean squared error. The correlation of residuals of the bivariate VAR is 50.74%, which helps to use one series’ more information
to forecast the other series’ unknown value.
This is real time forecast with revisions in data set. For each time period forecast, we collect
all the information available at that time to do one and two-step ahead forecasts. Figure 3 shows
revisions in data set and data set used for estimation in each time period. Knowing adp’s first
release is always two days prior to bls’s first release, we take advantage of two days’ ahead in adp
and forecast bls in the same time period. For instance, on May 30 of 2007, once ADP reports its
first release of nonfarm private employment for May (2007m5), the real time information at that
time is reported in column 2 and 3 in Figure 3, totally 76 observations. After the BLS reports
its employment situation for May on Jun 1 of 2007, bls on 2007m4 becomes the second revisions
and bls on 2007m3 becomes final revision. This can be seen from column 3 and 4 in Figure 3.
Furthermore, on July 4 of 2007, once ADP reports its first release of nonfarm private employment
for June (2007m6), the estimation period is from 2001m1 through 2007m5 of both adp and bls,
totally 77 observations. This continues till March of 2008. On April 4 of 2008, the BLS reports the
first release of employment situation for March (2007m3) while the BLS revises on annual basis. It
is on the column of 25 in Figure 3.
Figure 3: Revisions in Data
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!
" ! # # $! # # $! # #
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$ $! $ $
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The detailed estimation methodology is investigated in the following two-stage multistep VAR
forecast. At the first stage, we estimate the bivariate VAR over the period from 2001m1 through
2007m4. Following the maximum likelihood estimation by VAR, one step ahead residuals prediction
of both series is straitforward to be made. Then we regress the residuals of the bls on the residuals
11
of the adp. Since we observe the actual error term of the adp at time 2007m5, the best fitted
residual of the bls at time 2007m5 is the estimated coefficients times the actual error term of the
adp at time 2007m5.
At the second stage, we reestimate the bivariate VAR over the period from 2001m1 through
2007m5. One-step ahead fitted values of both series are predicted for time 2007m6. This is our
best fitted values of the bls and adp at time 2007m6. If we want to do further forecasts, we
reestimate the bivariate VAR over the period from 2001m1 through 2007m6. The fitted values of
both series are the best forecasts at time 2007m7, and so on. This dynamic forecast is based on
the real time information, that is the latest actual value of adp. As well the adp time series is
highly correlated with the bls time series, thus this two-stage multistep VAR forecast benefits from
accuracy measures.
4.3
Forecast Accuracy Comparison
Multistep ahead forecasts, computed by iterating forward the reduced form VAR, are assessed
in Table 2. The BLS releases revision of past employment announcements for the previous three
months, after which the announcement is considered final. To make our multistep ahead forecasts
consistent and comparable, we choose forecast horizon from 2007m5 through 2008m4. Because the
ultimate test of a forecasting model is its out-of-sample performance, Table 2 focuses on out-ofsample forecasts over the period from 2001m1 through 2007m4. It examines forecast horizons of
one month and two months. The dynamic forecast ‘k’ steps ahead is computed by estimating the
VAR through a given month, making the forecast ‘k’ steps ahead, reestimating the VAR through
the next month, making the next forecast and so on through the forecast period. The key difference
between the updating VAR forecasts and the ordinary VAR forecasts is the first stage estimation.
Taking advantage of one more information at the first forecasting period, the updating VAR forecast
provides the best estimate we are going to predict.
Table 2: Root Mean Squared Errors of Out-of-Sample Forecasts, 2001m1-2007m4
Forecast Horizon
1 month
2 months
†
Updating VAR
71.51
95.64
VAR
86.14
108.44
AR
71.75
96.44
The entries are root mean squared error.
As a comparison, out-of-sample forecasts were also computed for a univariate autoregression
with one lag, that is, a regression of the variable on lags of its own past values. Table 2 shows the
root mean square forecast error for each of the forecasting methods. The mean squared forecast
error is computed as the average squared value of the forecast error over the 2007m5 - 2008m4 outof-sample period, and the resulting square root is the root mean squared forecast error reported
12
in the table. In Table 2 the entries of column 2 through 5 are the root mean squared errors
of the updating VAR forecast, the ordinary VAR forecast, and univariate autoregression forecast
respectively for nonfarm payroll employment bls. The results indicate that the updating VAR
forecast has lower root mean squared error than the ordinary VAR and univariate autoregressive
forecast over one and two-step ahead forecast. Comparing to the ordinary VAR quantitatively, the
updating VAR forecast improves 17% in one-step ahead forecast and improves 12% in two-steps
ahead forecast.
One-Step Ahead Time Series Forecast Performance
One-step ahead time series forecast is based on time period of May 2007 through April 2008.
Totally 12 observations are constructed. For instance, on May 30 of 2007, after ADP reported the
first released nonfarm private employment, we use adp from 2001m1 through 2007m5 and bls from
2001m1 through 2007m4 to forecast bls on 2007m5, in which BLS will release it two days later on
June 1 of 2007.
−100
0
100
200
Figure 4: One-Step Ahead Forecast: 2007m5 - 2008m4
2007m4
2007m7
2007m10
Forecast Horizon
Actual Value
VAR Forecast
2008m1
2008m4
Updating VAR Forecast
Univariate Autoregression Forecast
Omitting adp data on 2007m5, the bivariate ordinary VAR forecast is to estimate both adp
and bls from 2001m1 through 2007m4. Then one-step ahead forecast is followed. The bivariate
updating VAR forecast takes two stages to complete forecasting. In the first stage, it estimates
both adp and bls through time period of 2001m1 to 2007m4. Since we know the true value of adp
on 2007m5, we have the true residual term of adp on 2007m5. In the second stage, we regress
residuals of bls on residuals of adp and obtain the coefficients of adp. Given the true residuals of
adp on 2007m5 and the coefficients of adp, the residual of bls on 2007m5 is obviously the outcome
of the true residual of adp on 2007m5 multiplied by the coefficients of adp. Then one-step ahead
forecast of bls is its fitted value adding estimated residual on 2007m5.
Consequently, standing at 2007m5, one-step ahead forecast of bls is based on both adp and bls
through time period of 2001m1 to 2007m5 and adp on time period of 2007m6. Keeping the process
13
to continue, we obtain 12 observations of one-step ahead forecasts in Figure 4. The solid line is
the BLS first released data from 2007m5 through 2008m4. The first released data plays a key role
in markets, so that we employ it as the actual value. We find that the updating VAR forecast
outperforms the ordinary VAR and univariate autoregression forecasts.
Two-Steps Ahead Time Series Forecast Performance
Two-steps ahead time series forecast is based on time period of June 2007 through April 2008.
Totally 11 observations are constructed. The difference from one-step ahead forecast is that based
on one-step forecast of bls on 2007m5, an iterated one-step forecast of bls on 2007m6 is conducted.
Since two-steps ahead forecast is based on one-step forecast, the updating multistep VAR forecast outperforms the ordinary VAR forecast. Figure 5 reports that the updating VAR forecast
outperforms the ordinary VAR and univariate autoregressive model forecasts.
−100
0
100
200
Figure 5: Two-Steps Ahead Forecast: 2007m6 - 2008m4
2007m4
2007m7
2007m10
Forecast Horizon
Actual Value
VAR Forecast
4.4
2008m1
2008m4
Updating VAR Forecast
Univariate Autoregression Forecast
Stock Market Reaction to Employment Announcement
Total nonfarm payroll employment by the BLS is the first major economic indicator released
each month. This is an economic report that can move the markets. Figure 6 gives the full picture
of quantitative comovement between the BLS employment situation first release and Don Jones
industrial IN. closing price. For instance, on Sept 7 of 2007 the BLS reported four thousands
decreases of employment. This led the stock market index declined dramatically by 250 points
at its closing price. On Nov 2 of 2007, the BLS reported 166 thousands increases in nonfarm
employment. This led the stock index closing price increased by 28 points at that day.
14
Figure 6: Impact of Report Release on Stock Market
†
Don Jones Industrial IN. from Yahoo Finance at http://finance.yahoo.com/
5
Conclusion Remarks
In multivariate time series, the covariance matrix of observation innovations plays an important
role in forecasting. We propose a practical method to update forecasts in multivariate VAR models.
The focus and the benefit of employing the updating approach is that the true innovation of
the current known observations is always to be useful to predict an innovation of the unknown
observations. The theoretical framework shows that the current known observations of one variable
are always going to be useful for forecasting the current unknown observations of another variables.
Therefore, higher correlation among observation innovations of multi-variables implies that mean
squared forecast error of the current unknown observations of the other variables will be accurate
for longer.
There are large amounts of applications in real-time forecasting. This paper uses U.S. labor data
to examine whether ADP estimates which is usually announced two days prior to BLS estimates
are helpful to forecast the same month total nonfarm payroll employment by the BLS. Rather
than use the final released data to estimate and forecast, we use data of as many different as there
are dates in the sample. More specifically, at every date within a sample, variables in the model
is the most up-to-date estimates at that time. We find that the predicted employment are more
accurately matching the labor data by considering the real-time information than the standard
time series forecast. Comparing to the ordinary VAR quantitatively, the updating VAR improves
17% in one-step ahead forecast and improves 12% in two-steps ahead forecast.
The timing of data release for time series variables for the same time period of observation
is often spread over weeks. For instance earning announcements for firms can be spread over a
15
two week period even though these earnings are for the same quarter or month. Future research
will extend the theoretical framework to more general case of two more variables available earlier
by public use. Applications to earnings forecasts for firms by different industry sectors are to be
studied.
16
Appendix
Proof of proposition 1.
For the forecast horizon at time T + 1, we observe ²1T +1 , since
²1T +1 = y1T +1 − (a11 y1T + a12 y2T ). So we can forecast the innovation ²2T (1) by the relationship
²2T (1) = ρ12 σσ12 ²1T +1 . Then forecast error becomes:
y2T +1 − ŷ2T +1 = ²2T (1)
The variance of the forecast error of y2 at T + 1 is
M SE[y2T (1)] = V ar[y2T +1 − ŷ2T +1 ] = V ar[²2T − ²2T (1)] = (1 − ρ212 )σ22
Proof of proposition 2.
Given information set I = {y11 , y12 , ..., y1T +1 , y21 , ..., y2T }, where the
first subscribe represents the variable and the second subscribe represents the time period. We
stand at T + 1. We know the time series of y1 from 1 till T + 1 while we only know the time series
of y2 from 1 till T .
At T + 1, we adopt all the time periods from 1 to T + 1 of y1 and the time periods from 1 to T
of y2 . Since ²1 and ²2 are correlated, we see ²2T +1 = (ρ12 σ2 /σ1 )²1T +1 . Thus the forecast error is
Ã
y1T +1 − ŷ1T +1
y2T +1 − ŷ2T +1
!
Ã
=
!
0
ε2T +1
Ã
=
!
0
(ρ12 σ2 /σ1 )ε1T +1
and the MSE or forecast error covariance matrix is
Ã
M SE[YT (1)] = V ar[YT +1 − ŶT +1 ] =
0
0
!
0 ρ212 σ22
At T + 2,
M SE[YT (2)] = M SE[YT (1)] + A1 Ω² (A1 )0
At T + 3,
M SE[YT (3)] = M SE[YT (2)] + A2 Ω² (A2 )0
Recursively, our updating forecast error covariance matrix becomes
Ã
u
M SE [YT (k)] =
0
0
0
ρ212 σ22
!
i
i 0
+ Σk−1
i=1 A Ω² (A )
Proof of proposition 3. Suppose we estimate the ordinary least squares model of the form
y2t = βy1t + ut
17
and the true DGP is as (5). OLS estimator (β̂ols ) is
β̂ols =
Σt y1t y2t
Σt y1t (αy1t−1 + βy1t + ut )
Σt y1t y1t−1 Σt y1t ut
=
=β+α
+
2
2
2
2 .
Σy1t
Σt y1t
Σt y1t
Σt y1t
Since y1t and ut are correlated, the second term goes to α in the limit while the third term does
not go to zero in the limit and the estimator is biased, that is,
E(β̂ols ) = β + α + E(
Since
α=
Σt y1t ut
2 ).
Σt y1t
V ar(y2t−1 )Cov(y1t−1 , y2t ) − Cov(y1t−1 , y2t−1 )Cov(y2t−1 , y2t )
=0
V ar(y1t−1 )V ar(y2t−1 ) − Cov(y1t−1 , y2t−1 )2
by linear regression of (7), we have
E(β̂ols ) = β + E(
Σt y1t ut
2 ).
Σt y1t
To see the inconsistency,
P
y1t (βy1t + αy1t−1 + ut )
P 2
)
y1t
P
y1t ut
β + plimn→∞ P 2
y1t
Cov(y1t , ut )
β+
V ar(y2t−1 + ε1t )
ρσ1 σ2 − βσ12
β+
σ22 + σ12
β
ρσ1 σ2
+ 2
2
2
1 + σ1 /σ2
σ1 + σ22
β + ρσ1 /σ2
.
1 + σ12 /σ22
plimn→∞ (β̂) = plimn→∞ (
=
=
=
=
=
Thus β̂ will underestimate β if β > ρσ2 /σ1 and will overestimate β if β < ρσ2 /σ1 . The degree of
underestimation or overestimation depends on σ12 /σ22 .
18
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